In ancient times, the only people who were offered a chance to become the priest were Aaron and his sons.

Would not this be unfair? There might have been people who were more fitted to the position. It has been shown throughout the Bible that sons of people with positions of priests were not always necessarily good (Eli's sons, Samuel's sons, etc.)

What was the reason for this strictness regarding the position of being the priest?

3 Answers
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Aaron was originally assigned to be Moses' spokesman to Pharaoh because Moses tried to argue with God over his inability to speak well (ironic, considering he had grown up in Pharaoh's house).

In keeping with many of God's directives, lineage was important to carry-on the traditions, types, and shadows of the Perfect One who would eventually come to rescue not only the nation of Israel, but openly make it possible for the world to be saved.

God also chose Judah to be the "host" tribe for the proper royal lineage of Christ. A quick glance at several of the descendants of David (we can even start with him) show some highly undesirable folks in the line to Christ - it doesn't make God's choice any less reasonable. (Indeed, the Israelites rebelled against God when they chose Saul as their first king because he was not of the tribe of Judah.)

Human sin and disobedience does not make God wrong - it makes humans wrong.

Sidebar: Interestingly, God chose the Levites to be the tribe set aside for duties related to worship. Levi and Simeon were the two brothers who killed every man in Shechem after the prince raped their sister Dinah. (Whether Dinah was at all complicit in the event is potentially debatable.)

What was the reason for this strictness regarding the position of
being the priest?

I have come to believe that most of the commandments from God have a logical backing. And one simply has to image herself being present in the situation and position of God to understand it.

Suppose you were setting up a software company. How would you go about hiring good programmers? The first thing might be to put an advertisement for people with a computer science degree. Now why this strictness about having a college degree? Though it is possible to find good programmers who don't have one, these cases are rare and exceptional. A degree though just a piece of paper shows that the person has put substantial time, effort and money in systematic study of the field. And since -

Practice makes a [wo]man perfect.

Hence our strictness.

The Levites were a tribe that had a special job. They were not supposed to take up other professions like agriculture, metalworking or trading etc. Their life was made up of these things -

Recite and memorize the scriptures.

Give sermons and practice oratory skills

Meticulously copy manuscripts by hand and double-triple check them for copy errors.

Clean up and maintain the temple.

Coordinate social ceremonies and religious celebrations.

This is not to say that others could not excel in these things but that the probability was very less. Much less than in modern times because of these reasons -

There were no schools, no books and no internet to gain information not relevant to your profession.

Life was much more busy. Cooking food on coal fire, cleaning up the barn and walking to the next village took entire day. This left little time for independent research for pursuing your ideal profession.

Now put yourself in the situation. You are setting up a temple. What would your advertisement read? Why your strictness?

Matthew 22:14 For many are eligible but few are qualified. (Monika's
Translation)

If your notice how the Levites were chosen as the priests, it actually goes back to Pharaoh and the plague that God sent to kill the firstborn of every family. Since those who had blood on the doorframe from the Paschal lamb were ‘redeemed’ from the death, every family owed 'the God of Moses' their firstborn. Not only so but as God also spared them of the first born of their cattle and also did not destroy their crops etc, the idea of all the firstborn or first-fruits as ‘belonging to God’ arose. Therefore as God brought them out of Egypt and into a land to be His people, Israel owed the first-born and first fruit of everything to God.

The Lord also said to Moses, “I have taken the Levites from among the Israelites in place of the first male offspring of every Israelite woman. The Levites are mine, for all the firstborn are mine. When I struck down all the firstborn in Egypt, I set apart for myself every firstborn in Israel, whether human or animal. They are to be mine. I am the Lord.” (NIV Numbers 3:11-13)

Possibly to make it more organized, and also to better foreshadow the future priesthood of Christ by the calling of God, under the symbol of high priest, who offered atoning sacrifices for sin. God ordained that in instead of selecting each firstborn the whole tribe of Levi would be set apart as dedicated to God. The dedication has nothing to do with personal holiness or fitness for the job, the dedication is by calling. The idea is that only God can create the Priesthood. Aron and Christ are the only two properly called priesthood heads.

You may wonder why I say ‘only two’ when we know Aron’s sons were included in the priesthood. Well that is because Aron made an imperfect shadow of the future reality in Christ. Since Aron was not good enough to be a good shadow of Christ, in that he had to die, many priests came after him, representing the same single office. This is the highest reason of the priesthood, to prefigure Christ.

The ‘calling’ of the priesthood can be seen here:

This is why he has to offer sacrifices for his own sins, as well as for the sins of the people. And no one takes this honor on himself, but he receives it when called by God, just as Aaron was. In the same way, Christ did not take on himself the glory of becoming a high priest. But God said to him, “You are my Son; today I have become your Father.” (NIV Hebrews 5:3-5)

The idea that the old priesthood was imperfect partly because ‘the priests died’ but Christ’s priesthood is superior, as he ‘lives forever’, representing the true Priesthood (or true High Priest for under him all believers are priests under him) can be seen here:

Jesus has become the guarantor of a better covenant. Now there have been many of those priests, since death prevented them from continuing in office; but because Jesus lives forever, he has a permanent priesthood. (NIV Hebrews 7:22-24)

Conclusion: So ‘Why were Aron and his son only ones eligible for being a Priest?’ Because the priesthood represents Christ’s priesthood and this can only be derived by the ‘calling of God’, not from the dedications of men. It is as a symbol of Christ ‘redeeming the firstborn’ who are every person who has his blood spread on the doorframes of their heart.